The axis maps keep the disc (round six, `pipeDisc6_all`)

Proved

J6Seg.pipeDisc6_all

by xuanji · Oct 20, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

The axis maps keep the disc (PipeDisc6).

The following holds (Lean statement, in the scope of module Solutions.J6PipeDisc2):

theorem J6Seg.pipeDisc6_all (P : Prm) (hP : P.Ok) : PipeDisc6 P

Context. PipeDisc6, part 2: one B₀ line keeps the disc, and the theorem

b0seg_disc: from FitSpec6 (‖B_i‖ ≤ 1/2 and the ≤ 9 error of B0seg) and 18 ≤ 2^p, every row of the B₀ line of a DiscL p line of length t_i has modulus ≤ 2^p (rows l ≥ s_i are 0);
pipeDisc6_all: ∀ P, P.Ok → PipeDisc6 P.

Formalization Note. Part of the Lean formalization of the round-six κ-improvement (integer-multiplication-kappa campaign, target IntMul.Kappa.jain_round6_framework). The statement is copied verbatim from the Lean development; the platform node carries the same declaration name. Definitions it mentions live in the IntMul_Kappa_R6_Defs* bundles (on top of the published κ bundles IntMul_Kappa_K_Defs*).

Preamble

-- Round-six κ development (integer-multiplication-kappa), generated by skeleton subtraction.

import Definitions.Def_IntMul_Kappa_R6_Defs63

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg
open Complex
open IntMul IntMul.HvdH IntMul.HvdH.Ref NLogN.KFM NLogN.KFY IntMul.Kappa.R6
open NLogN.KFB (GI gflat)

Formal statement

theorem J6Seg.pipeDisc6_all (P : Prm) (hP : P.Ok) : PipeDisc6 P := by sorry

Source

S. Jain, round-six networks for the integer-multiplication κ-improvement campaign (https://github.com/Swapnil-jain/integer-mult-kappa, commit 5868dc7) and D. Colkitt, integer-mult-bounds (https://github.com/CrocSwap/integer-mult-bounds); target `IntMul.Kappa.jain_round6_framework`. Lean declaration `J6Seg.pipeDisc6_all`, Solutions/J6PipeDisc2.lean (lines 66–164) of the round-six development.

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